Most every disability insurance contract does require you to file for social security disability if the insurance company asks you to. No matter which attorney you choose for the social security claim, you will be reimbursed by the disability company, so they are all "free" in that respect.
But here's the real problem. While the social security company representing you should be looking after your interests only, they often (most often--always, in my experience) are working aggressively to make sure the disability company is going to get its repayment of benefits. They'll have you sign a form, typically authorizing them to put their hands right into your checking account to grab money if you get your SSDI benefits.
In my opinion, that's a major conflict of interest.
Here's another thing I've seen. Knowing that the long-term disability insurance company will pay minimal benefits if your disability is mental illness related, the "free" attorney recommended by the disability insurance company will work to make your SSDI claim a "mental" claim. That's good for your disability company and bad for you.
It is, in my opinion, always better to hire a good social security disability attorney who is local to you and who will look out for your best interests only.